Liquid Level Control of Coupled-Tank System Using Fuzzy-Pid Controller

Liquid Level Control of Coupled-Tank System Using Fuzzy-Pid Controller

Trinh Luong Mien

Falculty of Electrical and Electronic Engineering University of Transport and Communications

No. 3 Cau Giay, Lang Thuong, Dong Da, Hanoi, Vietnam

Abstract: Liquid level control of coupled-tank is widely used in the chemical industry – the environment is often affected by noise. The article deals with the fuzzy-PID controller applied to the nonlinear dynamic model of the liquid level of the coupled- tank system, taking into account the effects of noise. Fuzzy-PID controller is designed based on PID initial parameters (determined based on the linear model) and fuzzy logic calculator for tunning PID parameters (suitable for nonlinear models and noise). The study results are caried out throught simulation model on Matlab using the coupled-tank nonlinear model with noise, applying the fuzzy-PID proposed controller, PID based on Ziegler Nichols.

Keywords: PID, Fuzzy, Level control, Coupled-tank

1. INTRODUCTION

   Liquid level control is always in great demand in the chemical industry, petrochemical refining, water treatment, power generation and construction material production. In these technological processes, the fluid is pumped, stored in a tank, and then pumped to another tank. Over the liquid is processed by chemical reaction and/or agitation in the tank, where the liquid level in the tank is controlled [1,2,14]. The coupled tank systems are commonly used in industries and the master of controlling the level of liquid in the tank, the flow control between the tanks is an important of all technological process control systems. Today's chemicals – the field has a tremendous impact on our economy [1,13,14]. Improving the quality of control and increasing the efficiency of the processing/production process is always required in this field, in order to reduce production/processing cost and lower production cost.

   Nonlinearity, associated kinetic and uncertainty are the major challenges posed by controlling the liquid level in the coupled-tank. Most of the coupled-tank object in published studies use a linear mathematical model when designing the controller, such as PID controlller [10,12], fuzzy controlller [9], fuzzy-PID [8], LQR, state feedback controller, model reference adaptive control [4,13].

   A number of recent studies have also addressed the nonlinear model of coupled tank using nonlinear control strategies such as sliding mode control [5,7,11], backstepping

   coupled-tank in the recently published works is good, but the implementation of these controllers is complex, the disturbance factor is not really considered.

   This article proposes a control approach: combining between the fuzzy logic calculator and traditional PID controller for a nonlinear model with noise of the liquid level coupled-tank control system. Firstly, the article presents a nonlinear model of the liquid level coupled-tank control system. Then, the PID controller is designed based on a linear model of the coupled-tank according to the method Ziegler- Nichols; designing the fuzzy logic calculator for tunning PID parameters applied in the nonlinear model with noise of the coupled-tank system. Finally, the study results is caried out throught simulation model on Matlab, showing the efiectiveness of the proposed control strategy.

2. DYNAMIC MODEL OF COUPLED-TANK SYSTEM This article deals with the coupled tanks with the two

   separate vertical tanks (see Fig. 1). Both tanks are interconnected by a flow channel where a rotary valve will be used to vary the sectional area of the channel by changing the discharge coefficient of the valve B. The liquid is fed into the first tank through the DC-motor controlled electric valve. Then the liquid flows to the second tank through the manual valve B, the liquid flows out of the tanks through the manual valve A or/and the manual valve C by adjusting the discharge coefficient of the valve A, C. The liquid level in the second tank is measured by the liquid level untrasonic sensor that converts the real physical level in the second tank l2 [cm] to an electrical voltage signal y [V].

   y ksl2 (1)

   where ks [V/cm] is gain of level untrasonic sensor.

   The control objective is to control the height of the liquid level in the second tank by manipulating the flow rate of the liquid into the first tank by means of the electric valve voltage. Assume that the valves output volume flow rate fi [cm3/s] is proportional to the manipulating voltage applied to electric valve u [V] as below equation:

   control [3], passivity based control [6], fuzzy logic controller [1], neuro-fuzzy-sliding mode controller [2]. It can be seen

   fi kvu

   (2)

   that the quality of the liquid level control system of the

   where kv is gain of the electric valve [cm3/s/V].

   Inlet

   u y

   MV,

   liquidkv

   Electric valve

   AC

   volltage signal

   Tank 1

   fi

   l1

   b, Cb

   CV,

   k

   voltage signal

   s

   Tank 2

   l

   Level untrasonic sensor

   Ve fi

   LC

   101

   LT

   l2SP

   LR

   101

   LAH

   101

   LI

   motor A1

   Pump

   fb

   Valve B

   2 A2

   l1

   A1 fb

   101

   A2

   l2

   101

   LAL

   101

   a, Ca

   Valve A Valve C

   fa fc

   c, Cc

   Va 1

   fa

   Vb 2 Vc

   fc

   Level Sensor

   Coupled- Tank

   (a). Schematic diagram of the coupled-tank apparatus

   (b). P&ID of the coupled-tank liquid level control system

   Electric Valve

   Level controller

   r u fi

   l2 y

   Voltage –

   Voltage

   Flow rate

   Level Voltage

   c). The block diagram of the coupled-tank control system

   Fig. 1. Description of the coupled tank liquid level control system

   The liquid used in the coupled tank is assumed to be

   dw1 w w w A dl1 f f f

   A dl1 f f f

   (6)

   steady, non-viscous, incompressible type of liquid. Applying Bernoulli's principle for the liquid at point 1 (before valve B)

   dt i a b

   1 dt

   i a b

   1 dt

   i a b

   and point 2 (after valve B) with corresponding pressure p1 and

   p2 (Fig. 1b), we have 2 cases:

   dw2 w w A dl2 f f dt b c 2 dt b c

   A dl2 f f

   2 dt b c

   (7)

   Case 1: when the liquid level in tank 1 is higher or equal the liquid level in tank 2, l1l2, the liquid flows from tank 1 into tank 2, we obtain the balance equation:

   where wi, wa, wb, wc are mass flow-rate; A1, A2 [cm2] are respectively section area of the first tank and second tank.

   p v2 p g(l l ) f bC 2

   Using above equations (2), (3c), (4), (5), thus we obtain

   1 2 2 1 2 b b f

   bC

   2g l l

   (3a)

   2

   2 b b 1 2

   dl 1

   where g=981[cm2/s] is acceleration of gravity; l

   [cm] is level

   1

   (kbu aCa

   2g l1 sign(l1 l2 )bCb

   2g | l1 l2 |)

   (8)

   3 1 3 dt A1

   in the first tank; [g/cm ] is liquid density; fb [cm /s] is volume flow rate through valve B, v2 [cm/s] is liquid velocity at point 2; b [cm2] is section area of valve B, Cb [%] is percentage of opening valve B

   dl2 1 (sign(l l )bC dt A 1 2 b

   2

   2g | l1 l2 | cCc

   2g l2 )

   (9)

   Case 2: when the height of the liquid level in tank 1 is less than in tank 2, such as l1<l2, the liquid flows from tank 2 into tank 1, we obtain the balance equation:

   The equations (8) and () represent a non-linear dynamic relationship of the liquid level (l1 and l2) in the two tanks with the ideal equations for the valves. In general applications, the square root law is only an approximation by solving directly

   fb bCb

   2g l2 l1

   (3b)

   the no-linear equations (8) & (9). But if the operating point is known and does not change quite often then it is convenient to

   Combining the above equations, we have flow-rate equation through the valve B as follows

   linearize the system obtained by first principles around the desired operating point. This makes the process significantly simpler and the model works well in a region around the

   fb sign(l1 l2 )bCb

   2g | l1 l2 |

   (3c)

   chosen operating point. This allows us to easily use linear control theory to design linear controller for the linear model of

   Similarly, we obtain the volume flow-rate equations

   through the valve A, C as:

   the coupled-tank, such as PID controller.

   fa aCa 2g l1

   f cC 2g l

   (4)

   (5)

   The linear model of the coupled-tank: At the desired operating point of the fluid level in the second tank L2s, the control system is at steady state, so on:

   c c 2

   where a, c [cm2] are respectively section area of valve A, C;

   dL1s 0 1 (k U aC

   2g L

   • sign(L

   • L )bC

     2g | L L |)

     (10)

     and Ca, Cc [%] are percentage of opening valve A, C

     dt A1

     b s a 1s

     1s 2s b

     1s 2s

     respectively.

     dL2s 0 1 (sign(L

   • L )bC

     2g | L L

     |) cC

     2g L )

     (11)

     The coupled tank dynamics are based on the principle of

     dt A2

     1s 2s b

     1s 2s c 2s

     mass balance which states that the rate of change of liquid mass in each tank equals the net of liquid mass flows into the tank. Here it assumes that the liquid density and cross area of tanks are constant

     where L1s is height at steady state, Us is pump voltage at at steady state.

     DA	Inner diameter of valve A	0.5	cm
     DB	Inner diameter of valve B	0.7	cm
     DC	Inner diameter of valve C	0.5	cm
     Hmax	Max. height of liquid level in tank 1, 2	30	cm

     Considering a small incremental change in the control input, u in Us, which subsequently cause an incremental change in height in the two tanks, l1 in L1s and l2 in L2s. Hence, equations (8) and (9), assuming that the fluid always flows from tank 1 to tank 2, can be re-written as:

     d(l1 L1s ) 1 [k (u U ) aC

     2g l L bC

     2g l l L L ]

     (12)

     Assume that the desired height of fluid level in the second

     dt A1

     b s a

     1 1s b

     1 2 1s 2s

     tank L2s

     =15[cm], from equations (10), (11) and (20), we obtain

     d(l L ) 1

     the linear model of the liquid level process of the coupled-tank

     2 2s

     (bC

     2g l l L L

   • cC

   2g l L )

   (13)

   system as below:

   b

   dt A2

   1 2 1s

   2s c

   2 2s

   G (s) l2 (s) 0.0176

   L2

   (21)

   Following Newton's binomial generalized theorem, if x<<1 then we can approximate:

   u(s)

   s2 0.362s 0.007

   (1 x) 1 x

   (14)

3. THE FUZZY-PID CONTROLLER DESIGN FOR

   Applying the above approximation (14), we obtain the below equations:

   LIQUID LEVEL PROCESS OF THE COUPLED- TANKS SYSTEM

   l1 L1s

   L (1 l1 )

   1s

   L1s

   l

   L (1 l1 )

   1s

   2L1s

   l

   l1

   L

   2 L

   1s

   1s

   l

   (15)

   The structure of the fuzzy-PID controller for the liquid

   level process of the coupled-tank system is proposed as in Fig.

   2. The fuzzy-PID controller is a combination of the basic PID and the fuzzy logic calculator. The initial parametters

   l2 L2s

   L2s (1 2 )

   L2s (1 2 )

   L2s 2

   (16)

   kP0 , kI 0 , kD0

   of the basic PID are definited based on the

   L2s

   2L2s

   2 L2s

   common methods, such as Ziegler Nichols (PID-ZN), Chien-

   l l L L

   (L L )(1 l1 l2 )

   L L

   l1 l2

   (17)

   Hrones- Reswick (PID-CHR). The

   kPF , kIF , kDF

   are seft-

   1 2 1s 2s

   1s 2s

   L1s L2s

   1s 2s

   2 L1s L2s

   tunning parametters of PID based on fuzzy logic calcutalor (FuzzyCal block in Fig.2) for the nonlinear model of coupled-

   Substitute these approximation equations (15-17) into

   (12-13) and in combination with equations (10-11), we obtain:

   dl1 (k k )l k l kb u (18)

   dt 1 2 1 2 2 A

   1

   dl2 k l (k k )l (19)

   tank with the noise.

   1. Designing the basic PID controller

      The basic PID is designed based on the linear model of the liquid level process of the coupled-tank system. Using the Ziegler Nichols method, we can determine the initial

      dt 3 1 3 4 2

      paramaters kP0

      , kI 0

      , kD0 .

      k aCa g ; k bCb

      g ; k bCb g ; k cCc g

      The transfer function of the level control object as:

      1 A 2L

      2 A 2(L L ) 3 A 2(L L ) 4 A 2L

      1 1s

      1. 1s 2s

      2. 1s 2s

      2 2s

      The equations (18) and (19) describe the linear model of the coupled-tank system, where input is the incremetal pump voltage u(t) , and output is the incremetal fuild level in the

      Gobj (s) ksGL2

      (s)

      0.1074 K

      1 2

      s2 0.362s 0.007 (T s 1)(T s 1)

      (22)

      second tank l2 (t) . By taking the Laplace transform of equations (18-19) the following transfer function is obtained:

      where K=15.372, T1=2.93, T2=48.78

      Arcoding to the Ziegler Nichols 1st method, the parameters kP0 , kI 0 , kD0 can be determined as follows:

      GL2

      (s) l2 (s)

      u(s)

      s2 (k k

      k3kb / A1

      • k k )s (k k

      • k k

      • k k )

        (20)

        G s k

        • kI 0 k s

      (23)

      1 2 3 4 1 3 1 4 2 4

      In this paper, we design a fuzzy-PID controller applied for coupled-tank system with following parameters [15].

      where:

      C P0 s D0

      Tab 1. Constants involved in coupled-tank system of Fig. 1

      kP0

      1.2T2

      KT1

      1.2 * 48.78

      15.372 * 2.93

      1.31

      Parameter	Desctiption	Value	Unit
      kv	Gain of DC-motor electric valve	3.3	cm3/s/V
      ks	Gain of level untrasonic sensor	6.1	cm/s
      Ca	Percentage of opening valve A	60	%
      Cb	Percentage of opening valve B	80	%
      Cc	Percentage of opening valve C	60	%
      D1, D2	Inner diameter of tank 1, 2	6	cm

      k kP0

      I 0 2T

      1

      0.22

      T1

      kD0 kP0 2 1.92

      Howerver with the nonlinear model of the coupled-tank, the acceptable pamameters are kP0 = 15, kI 0 = 0.3, kD0 = 11.

      kP0

      kPF

      kI0

      kIF

      kD0

      de kDF

      dt

      Fuzzy Cal

      e

      1

      s

      r u

      Fuzzy-PID controller

      de

      –

      y

      Level Sensor

      Coupled- Tank

      Electric Valve

      p/>

      Level process of coupled-tank

      dt

      Fig. 2. Structure of fuzzy-PID for liquid level process of coupled-tank

   2. Designing the fuzzy logic calculator

   The fuzzy logic calculations block (FC) have: two inputs – level error in the second tank (EL), derivative of level error (DEL) corresponding to input voltage error signal e=y-r (r- level setpoint, y- level in tank 2) and de/dt; three output is PL, IL, DL corresponding to the output value kPF, kIF, kDF.

   Using membership functions are shaped triangular for all variables, fuzzied for all input variables by 5 fuzzy sets

   {NL (Negative Large), NS (Negative Small), ZE (ZEro), PS (Positive Small), PL (Positive Large)}, fuzzied for all output variables by 5 fuzzy sets {SM (SMall), ME (MEdium), LA (LArge), QL (Quite Large), VL (Very Large)}. The physical

   Tab.3. The basic fuzzy rule of kPL, kIL, kDL

   PL IL DL		EL
   		NL	NS	ZE	PS	PL
   DEL	NL	SM	SM	SM	SM	SM
   	NS	SM	ME	SM	SM	SM
   	ZE	SM	SM	LA	LA	QL
   	PS	SM	SM	LA	QL	VL
   	PL	SM	SM	QL	VL	VL

   Using the Max-Min composition rule and the cetroid defuzzification method, we can obtain the clear output value of FC: kPF, kIF, kDF for level control loop. Thus, the fuzzy- PID controller can be calculated by equations:

   I I 0

   D D0

   domain of the input & output variables are determined as: EL[-20,20], DEL[-2,2], PL[0,20], IL[0,1],

   *

   k = k

   P P0

   + kPF

   , k* = k

   + kIF

   , k* = k

   + kDF

   (24)

   DL[0,15].

   Depending on the characteristics of the level control proces of the coupled-tank and the PID control principle in order to improve quality control for this system (see Tab.2),

4. SIMULATION RESULT

   The simulated diagram of the fluid level process of coupled-tank system is described as Fig. 3. The fuzzy-PID controller is a combination of the FuzzyCal block with the VariablePID. The self-tunning parameters of fuzzy-PID is

   we define the 25 basic fuzzy rules as Tab.3.

   determine on equation (24), here

   kP0 , kI 0 , kD0

   are initial

   Closed-loop respond	Rise time	Steady time	Over- shoot	Steady error	Stability
   Increasing kP	Decrease	Small change	Increase	Decrease	Degrade
   Increasing kI	Small decrease	Decrease	Increase	Eliminate	Degrade
   Increasing kD	Small decrease	Decrease	Small decrease	Small change	Increase

   Tab.2. The effect of kP, kI, kD tunning

   parmaters of PID and kPF , kIF , kDF

   are the clear output value

   of the FuzzyCal block. The fluid level process of coupled- tank is used as nonlinear model, using equations (8) and (9).

   The simulation is carried out with three controllers: Fuzzy-PID, PID-ZN1, PID-CHR. The quality control system is evaluated through four indexes (overshoot, rise time, steady time, steady error) in two circumstances: (a). varying setpoint level; (b). as impacted by the bound noise with small margin.

   Fig. 3. Simulation of the fluid level coupled-tank control system using fuzzy-PID

   30

   L2 [cm]

   25

   20

   15 Setpoint

   PID-ZN1

   PID-CHR

   10 Fuzzy-PID

   5

   0

   0 20 40 60 80 100 120 140 160 180 200

   Fig. 4. Response curves of the level controllers as varying setpoint level

   Fig. 5. Response curves of the level controllers as having noise with small margin

   The simulation results, as using PID-ZN1, PID-CHR and Fuzzy-PID controller, is presented in Tab. 4.

   Tab. 4. Performance of Fuzzy-PID controller & others

   Controller Index	Fuzzy-PID	PID-ZN1	PID-CHR
   Rise time	Small, ~1.5s	Large, ~4.1s	Very large, ~8.4s
   Steady time	Small, ~3.1s	Large, ~11.2s	Very large, ~8.4s
   Overshoot	Not	Large, ~19.4%	Small, ~12.3%
   Steady error	Eliminate, or very small with noise	Very small, but large swing with noise	Very small, but swing with noise

   The simulating results show that fuzzy-PID has the best control quality: not overshoot, eliminating steady error, the smallest steady time and eliminating neraly the effect of the disturbaces, when it was compared to traditional PID controllers.

5. CONCLUSION

This paper has presented a case study where the basic PID controller is combined with the fuzzy logic calculator for the nonlinear model of the liquid level of coupled-tank system. The simulation results suggest that the fuzzy-PID proposed controller can be applied to the liquid level control process in the chemical industry, where noise is always presented. The fuzzy-PID controller can improve quality of the liquid level coupled-tank control system, increase the process efficiency and bring economic benefit to end-user. However, we need to study in more detail about dynamics of actuator & sensor, according to the actual device to obtain a more realistic control object model, which helps to control the fluid level in coupled-tank better.

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Trinh Luong Mien obtained his PhD degree in automation and control of technological processes and manufactures at Moscow State University of Railway Engineering (MIIT) in Russia Federation in 2012. Trinh Luong Mien is a lecturer at Faculty of Electrical and Electronic Engineering, University Transport

and Communications in Vietnam since 2004. His main research is the development of intelligent control algorithms for the technological and manufacturing processes in industry and transportation based on fuzzy logic, neuron network, adaptive & optimal theory; study algorithms controlling & ensuring the safe movement of the electrical train in ATP/ATO/ATS/ATC system of the urban railway; design the supervisory control system based on IoT platform.